Chop or tear the lettuce into a salad bowl. Add almonds, walnuts, and soy nuts, then toss.
Mix the lime juice and water in a small bowl. Add vegan sugar to taste, or omit it. Stir in the sesame seeds.
Pour a little lime juice into a baggie and put the apple chunks in it, shut the baggie, and work the lime juice around until all the apples are coated (it keeps them from getting all brown and nasty).
Take the apples out and add them to the salad bowl. Toss, add 1 tablespoon lime juice, and toss again. Leave the lime juice-water-sesame seed mixture on the side or drizzle it on as dressing.